The foreign secretary said action was needed in the areas most affected by Covid-19 to avoid a national lockdown, and he urged Burnham to relent.
"We can't have a situation where Andy Burnham is effectively saying unless you give us what we want, we're not going to do the right thing in terms of following the new rules, which will protect the very people of Manchester he's elected to represent," Raab said in an interview with BBC Breakfast.
Government talks with leaders in Lancashire continued on Friday morning about moving the region into the highest measures of lockdown.
The dispute between ministers and northern leaders including Burnham centres on the financial package on offer for people whose livelihoods are affected by closures.
On Thursday, MPs from Greater Manchester and Lancashire backed the mayor in emphatically rejecting the tier 3 move. Speaking to media outside Manchester's central library, Burnham said he would not accept the area being treated "as canaries in the coalmine for an experimental lockdown strategy", which he said not even the government's medical advisers thought would work.
In response to Raab's comments on Friday, Burnham tweeted: "It's not about what we want for ourselves, Dominic Raab. It's about what we want for low-paid and self-employed people everywhere: fairness."

But he said the feeling among experts was that placing only areas with the highest cases into the strictest measures might not be successful, instead proposing a series of two-week "circuit breaker" national lockdowns.
"This three-week delay between the impact of interventions and it affecting hospitalisations I think is causing considerable confusion and means that we're already going to be very close to where we were in April within two or three weeks," he told BBC Radio 4's Today programme.
While Raab admitted on Today that there was a risk that the government's new tiered approach to local lockdowns would not work, he said: "The answer to that risk is to lean in, make it work as effectively and as swiftly as possible."
He praised London's mayor, Sadiq Khan, ahead of the capital heading into tier 2 restrictions, and the Liverpool city region's mayor, Steve Rotheram, after tier 3 measures were imposed on the area on Wednesday, for working with the government, adding that the "door will always be open for Andy to do so".
Raab said ministers would continue to work with Greater Manchester leaders to reach an decision, but he told Sky News that "obviously in the last resort, the government has the powers to proceed [with imposing tier 3 restrictions] in any event."
Manchester city council's leader, Sir Richard Leese, told Sky News that discussions had stalled, with no further meeting planned between ministers and Greater Manchester leaders on Friday.
Comment: A similar situation played out in Madrid, where the local government rejected the destructive and provably ineffective lockdown measures, it even won its case in course, but then central government simply smacked Madrid into 'state of emergency', thus wrenching any control the local leaders had.
Local leaders throughout Europe have rejected the lockdown measures and yet, for some inexplicable reason, central governments, claiming to be acting independently meanwhile imposing exactly the same restrictions as each other, within days of each other, disregard their pleas and railroad over any ideas of democracy, and logic.
Here's what another local leader from the north of England had to say about the local lockdown:
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